First-Principles Calculation of the Elastic Moduli of Sheet Silicates and Their Application to Shale Anisotropy

American Mineralogist, Volume 96, pages 125–137, 2011

First-principles calculation of the elastic moduli of sheet silicates and their application to shale anisotropy

B. Mi l i t z e r,1,2,* H.-R. We n k ,1 S. St a c k h o u s e ,1 a n d L. St i x r u d e 3

1Department of Earth and Planetary Science, University of California, Berkeley, California 94720, U.S.A. 2Department of Astronomy, University of California, Berkeley, California 94720, U.S.A. 3Department of Earth Sciences, University College London, Gower Street, London WC1E 6BT, U.K.

Abs t r a c t The full elastic tensors of the sheet silicates muscovite, illite-smectite, kaolinite, dickite, and nacrite have been derived with first-principles calculations based on density functional theory. For muscovite, there is excellent agreement between calculated properties and experimental results. The influence of cation disorder was investigated and found to be minimal. On the other hand, stacking disorder is found to be of some relevance for kaolin minerals. The corresponding single-crystal seismic wave velocities were also derived for each phase. These revealed that kaolin minerals exhibit a distinct type of seismic anisotropy, which we relate to hydrogen bonding. The elastic properties of a shale aggregate was predicted by averaging the calculated properties of the contributing mineral phases over their orientation distributions. Calculated elastic properties display higher stiffness and lower p-wave anisotropy. The difference is likely due to the presence of oriented flattened pores in natural samples that are not taken into account in the averaging. Keywords: Elasticity, clay, ab initio calculations, sheet silicates, seismic anisotropy

Introduction the case of shales. One reason is that porosity and grain contacts Sheet silicates are among the minerals with highest elastic are inadequately accounted for. Another reason is the uncertainty anisotropy. Aggregates containing oriented sheet silicates, such in the single-crystal elastic properties of sheet silicates (Chen as schists and shales, also display very high anisotropies and and Evans 2006). this has important implications for interpreting seismic travel Despite their importance, the single-crystal elastic moduli of times through such rocks. Shales are the most abundant rocks in clay minerals have not been measured experimentally, because of sedimentary basins and relevant for petroleum deposits (e.g., Jo- technical difficulties associated with the small grain size. There hansen et al. 2004) as well as for carbon sequestration (Chadwick is also considerable uncertainty in atomic force microscopy et al. 2004). Their physical properties are of great importance (Prasad et al. 2002). For illite, elastic constants of the analog in exploration geophysics (e.g., Mavko et al. 1998; Wang et al. mineral, muscovite, have traditionally been used (Vaughan and 2001). Elastic properties of shales have been measured with Guggenheim 1986; Zhang et al. 2009) but considerable uncer- ultrasound techniques (e.g., Hornby 1998; Hornby et al. 1994; tainty arises from using such analogies. For kaolinite, values Johnston and Christensen 1995), because of their importance for from first-principles calculations exist for the ideal structure seismic prospecting of hydrocarbon deposits (e.g., Banik 1984). (Sato et al. 2005; Mercier and Le Page 2008). However, the In metals and igneous and metamorphic rocks, anisotropic ag- structure of kaolin minerals in natural samples is expected to gregate properties can be satisfactorily predicted by averaging the differ considerably from the ideal kaolinite. Moreover, other single-crystal elastic properties over the orientation distribution equally important sheet silicates have not yet been examined (Bunge 1985; Barruol and Kern 1996; Mainprice and Humbert by experiment or theory, including illite-smectite. In view of 1994). In the case of shale, anisotropy can be calculated with this, in the present work, we have calculated the single-crystal this method, but absolute values of predicted elastic stiffness elastic properties of illite-smectite and the kaolin minerals kao- coefficients are over a factor of two higher than the measured linite, dickite, and nacrite, which have structures more similar results (Valcke et al. 2006; Voltolini et al. 2009; Wenk et al. 2008). to those found in nature. In addition, to establish and test our Currently only empirical models are available to describe the method, we have computed the elastic properties of muscovite, elasticity of shales that do not rely on measured microstructural for which analogous experimental data exists. The computed properties (e.g., Bayuk et al. 2007; Ougier-Simoni et al. 2008; elastic properties are then used to predict the elastic properties Ponte Castaneda and Willis 1995; Sayers 1994). There are vari- of the classic Kimmeridge Clay shale (age 151–156 Ma) from ous reasons why the simple averaging schemes are limited in Hornby (1998) for which experimental elastic properties are available, as well as orientation distributions of component * E-mail: [email protected] phases (Wenk et al. 2010).

0003-004X/11/0001–125$05.00/DOI: 10.2138/am.2011.3558 125 126 Militzer et al.: Ab initio Calculation of Elastic Moduli of Sheet Silicates

→ → → Wenk 2009) with an orthogonal coordinate system defined by Z||→c, Y||→c×a→, and X Co m p u t a t i o n a l m e t h o d s → → → = Y×Z and (2) a setting where the Z is perpendicular to the tetrahedral planes (001) We used the Vienna ab initio simulation package (VASP) (Kresse and Hafner → → → → → → → → in both monoclinic and triclinic crystal structures: X||a, Z||a×b, and Y = Z×X. The 1993; Kresse and Furthmüller 1996) to perform density functional calculations latter choice preserves the direction of C33 when comparing different stackings in using either the local density approximation (LDA) or the → Perdew-Burke-Ernzerhof kaolinite polytypes that have different c axes. (Perdew et al. 1996) generalized gradient approximation (GGA) in combination with the projector augmented-wave method (Blöchl 1994). The wave functions of the valence electrons were expanded in a plane-wave basis with an energy cut-off Mo d e l s y s t e ms of 586 eV or higher and the Brillouin zone was sampled using 4 × 2 × 2 and 6 × 4 × 4 k-points grids (Monkhorst and Pack 1976) depending on the size of the unit Muscovite –3 cell. All structures were relaxed until all forces were <10 eV/A. Muscovite KAl2[AlSi3O10](OH)2 is a 2:1 sheet silicate, with To calculate the elastic constants we used the finite strain method (Karki et al. → ↔ each layer comprising an octahedral sheet fused between two 1997, 1997b, 2001). The lattice vectors of the strained unit cell, (a→′, b′, →c′) ≡ A′, → ↔ ↔ ↔ ↔ ↔ were obtained from the unstrained cell, (a→, b, →c) ≡ A, using A′ = (1 + ε) A, where tetrahedral sheets (Fig. 1). Due to isomorphic substitution, it ↔ ↔ 1 is the unit matrix and ε is the strain tensor. The diagonal and off-diagonal strain exhibits structural disorder in the tetrahedral sheets (Collins and tensors were of the form (Ravindran et al. 1998) Catlow 1992), with 25% of sites occupied by Al3+ cations and 75% by Si4+ cations. This leads to the layers having a net nega-      δ 00  000  tive charge, which is balanced by K+ anions residing between    ;    , ε =    1  000 ε4 = 00δ / 2  the layers.      000  02δ / 0  Since it is possible that cation disorder will affect the elastic properties of the phase, we performed first-principles calculations where the indices are given in Voigt notation. The remaining strain tensors are to determine the most favorable arrangements. obtained by index permutations. For each strain, we performed two calculations 4+ 3+ using δ = ± 0.005 and determined the corresponding stresses. The elastic constants Isomorphic substitution of Si by Al , in the tetrahedral 3+ were then determined from simple stress-strain relations. Consistent results were sheets, creates a local negative charge in the region of the Al also obtained for other values of δ. Mainprice et al. (2008) performed similar cation. It is therefore reasonable to assume that Al3+ cations calculations for talc. within the same tetrahedral layer, will tend to repel one another. The error associated with our computed elastic constants, due to controlled Our calculations support this. For an 84 atom unit cell, placing ± approximations, is estimated to be < 2 GPa, with the major part of this associated 3+ with the careful, but slightly imperfect, structural relaxation and use of a finite two Al cations in the same tetrahedral sheet increases the energy 3+ strain. In addition to this, we must also take into account the uncontrolled ap- by 1–2 eV, while placing all four Al cations in the same sheet proximation of adopting a specific exchange-correlation functional. While LDA increases it by 7 eV. Calculations for heterogeneous distributions and GGA have been shown to predict many properties of a variety of materials in larger super cells showed a similar trend. This provides a strong with remarkable accuracy, neither approach is perfect. LDA tends to overbind the 3+ P=0 argument for having similar Al cation concentrations in each sheets in sheet silicates, and zero-pressure densities predicted with LDA (ρLDA) are too high (by 3% for muscovite). In contrast, GGA tends to underbind the layer, to reduce unfavorable interactions between them. P=0 sheets, and zero-pressure densities predicted with GGA (ρGGA) are too low (by If one considers the conventional unit cell, assuming an equal 5% for muscovite). Since the computed elastic constants are affected by changes Al3+ concentration in each layer, there is only one Al3+ cation to in density, we can improve the accuracy of the predicted constants by adopting distribute among four sites. Every site is energetically equivalent ρP=0 the experimental density ( EXP) and relaxing the structure at fixed volume. In this 4+ 3+ approach it is important to account for uncertainties in the volume, structure, and and leads to the formation of a Si -Al chain parallel to the b composition of the experimental sample and differences between it and the model lattice vector (Fig. 2a). However, a more favorable cation dis- system. By adopting the experimental value of the volume, we remove one of the tribution is obtained if one considers a simulation cell compris- main sources of error in elastic constants computed with DFT. ing two unit cells (Fig. 2b). This allows Al3+ ions to be placed A similar strategy was adopted by White et al. (2009) who studied the geometry 3+ 3+ of ideal kaolinite, comparing predictions for several different functionals, but with across the silicate rings and thus decrease Al -Al interactions. the lattice parameters fixed at the experimental values. Following the suggestion Monte Carlo simulations using larger super cells, with up to 36 of one of the referees, we investigated the difference between the elastic constants substitution sites, found the same arrangement of cations to be calculated with LDA, keeping all lattice parameters fixed at their experimental most favorable. In these calculations the optimization criteria was value, and using the cell parameters obtained by optimizing the cell at the experi- to maximize the distance between the Al3+ sites or alternatively mental density. The resulting elastic constants were found to be very similar, which is not surprising, because the lattice parameters obtained from optimization at the minimize the Ewald energy. experimental volume agreed well with the experimental values. However, doubling the size of the simulation cell would Imposing the experimental density affects the interlayer separations, but leads make first-principles calculations of the elastic constants of to little change in the geometry of the sheet itself. In clays, bonding between layers muscovite prohibitively expensive. We consequently used the is rather weak, which is the reason why it is difficult to describe it with density functional methods. Consequently, at the zero pressure densities, we find significant configuration shown in Figure 2a for all calculations of elastic differences in the LDA and GGA predictions of elastic behavior in the direction properties. Even for this configuration, there is still considerable perpendicular to the sheets, while that within the sheet is less affected. Karki et structural freedom, because one can choose a different site for al. (2001) have shown that LDA leads to good agreement with experiment for a the Al3+ cation in each layer. However the energy differences wide variety of mantle phases. In view of this, while we will compare LDA and P=0 arising from such permutations is relatively small (~0.07 eV per GGA results in the paper, our preferred elastic constants come from LDA at ρEXP. It should also be noted that we compute isothermal elastic constants at 0 K, unit cell) compared to that possible with different concentrations whereas experimental studies determine adiabatic elastic constants at room tem- in the layers (1–2 eV), because of the increased separation and perature. However we expect the difference between the two to be small and direct screening of intermediate layers. comparison between theoretical and experimental values to be reasonable. Out of the 44 ways to distribute four Al3+ ions equally among Optimized structures, for the different phases, are available as CIF files at the Inorganic Structural Database. To facilitate comparison with previous results, the four layers, we selected eleven configurations that either 3+ elastic constants are reported for two crystallographic settings: (1) The conventional had a very low Ewald energy or the Al ion separations were setting (e.g., Standards on Piezoelectric Crystals 1949; Nye 1959; Matthies and especially large. For those eleven geometries, we relaxed the MILITZEr ET AL.: AB InITIO CALCULATIOn OF ELASTIC MODULI OF SHEET SILICATES 127

P=0 atomic positions and unit cell with GGA at ρGGA. Six of the eleven relaxed structures were low in energy, within 0.07 eV per unit cell, which is a range that one would expect for a disordered mineral. For all six structures, we obtained very similar elastic constants, reported in Table 1. This leads us to conclude that cation disorder has very little impact on the elastic properties of muscovite. The largest deviations are seen in model 4 (Fig. 2), which has lower elastic constants mainly because the relaxation at zero-pressure led to a slightly lower density.

Illite-smectite Various structural models have been proposed for illite- smectite minerals (Moore and reynolds 1989; Sakharov et al. 1999; Plançon et al. 1985; Drits et al. 1994; Watanabe 1988). They contain mixed layers of illite and swelling layers of smectite that cause systematic peak shifts in X-ray powder patterns. The interstratification of illite and smectite changes from random alternation to full segregation. Contrary to muscovite, illite- smectites exhibit a wide range of compositions with significant na, Ca, and Mg interlayers. Their exploration goes beyond the scope of this work; instead we adopted an idealized structural model put forth by Stixrude and Peacor (2002).

Ta b l e 1. Elastic constants of muscovite calculated at zero pressure → → with GGA, in the conventional setting, Z||c Model 1 2 3 4 5 6

E-E1 (eV) 0 0.007 0.016 0.042 0.054 0.072 ρ (g/cm3) 2.682 2.683 2.688 2.675 2.690 2.689

C11 172.7 172.8 173.1 167.9 169.7 170.6 C22 166.7 166.9 167.7 158.6 167.0 165.6 C33 54.8 54.8 55.1 53.0 55.0 54.7 C44 14.2 14.3 14.7 9.4 14.9 15.2 C55 17.2 17.3 17.4 15.5 16.8 16.5 C66 67.6 67.6 67.8 66.5 67.5 67.5 C12 48.8 48.8 49.2 43.8 48.0 48.6 C13 20.1 20.2 20.4 17.7 20.6 20.5 C15 –17.7 –17.7 –17.6 –18.4 –17.3 –16.7 C23 17.4 17.5 17.9 14.2 18.0 18.5 C25 –1.7 –1.7 –1.7 –4.2 –1.8 –1.5 C35 –3.3 –3.2 –3.2 –4.9 –3.0 –2.5 C46 –4.4 –4.4 –4.5 –5.0 –4.1 –4.0 Notes: Each column corresponds to a model with a different tetrahedral Al3+/Si4+ cation arrangement. The energy differences with respect to model 1 are also given on line 2 in eV per 84 atom unit cell. One can see that cation disorder has little effect on the elastic properties of the phase.

Fi g u r e 1. Muscovite. Comparison of Al3+ disorder models 1 and 4 reported in Table 1. The light and dark tetrahedra represent the Al3+ Fi g u r e 2. Sheets of tetrahedral cations in muscovite. The dark and and Si4+ cation locations, respectively. The structure has been extended light spheres indicate the Al3+ and Si4+ ions, respectively. The left panel beyond the unit cell (marked by thin lines) for clarity but the Al3+:Si4+ shows the lowest energy confi guration for a 1:3 Al:Si ratio within the ratio is the same in both structures. The spheres denote K+ ions and the conventional unit cell (rectangular box). On the right, the lowest energy octahedra mark the remaining Al3+ cations. confi guration among all unit cells. 128 MILITZEr ET AL.: AB InITIO CALCULATIOn OF ELASTIC MODULI OF SHEET SILICATES

Illite-smectites are similar in structure to muscovite, in that Kaolin minerals they are composed of aluminosilicate layers, formed from an Kaolinite is a 1:1 sheet silicate, i.e., tetrahedral layers are only octahedral sheet fused between two tetrahedral sheets (2:1 sheet on one side of the octahedral sheet. Its ideal structural form is silicate; Fig. 3), however, the arrangement of cations in the tetra- triclinic with two Al2Si2O5(OH)4 formula units per unit cell and hedral layers is quite distinct. Just as one would assume from their a simple 1M stacking sequence (Bish 1993; neder et al. 1999) name, illite-smectites are interstratified minerals that comprise that is shown in Figures 3 and 4. However, natural kaolinites, in alternate layers of illite and smectite. In the present work, we sedimentary rocks exhibit considerable stacking polytypism and investigate one of the simplest illite-smectites known as rectorite, disorder (Bailey 1963, 1988; Kogure et al. 2002, 2005). Stack- where the illite layer has a structure similar to muscovite and ing polytypism of hydrous sheet silicates has been analyzed in the smectite layer has the pyrophyllite structure. Two types of detail by Bailey et al. (1988) who identified 12 standard types. interstratification have been considered by Stixrude and Peacor Stacking polytypism in kaolin minerals has also been investigated (2002): layer centered (model A) and interlayer centered (model by Zvyagin and Drits (1996). Artioli et al. (1995) went beyond B). In model A, alternate layers of illite and smectite, identical to the 12 standard types by developing a random stacking model, those found in the mineral end-members, are stacked in sequence. which significantly improved agreement with powder diffraction However in model B, each layer is identical and comprises one patterns of natural kaolinites. Dera et al. (2003) studied pressure- aluminum-poor and one aluminum-rich tetrahedral sheet; the up- induced changes in the stacking behavior. A comprehensive down alternation of these layers leads to two different interlayer recent review of the stacking polytypism in kaolin minerals environments: one enclosed by low-charge tetrahedral sheets is given by Mercier and Le Page (2008), who also provide an and the other by high-charge tetrahedral sheets, where potassium extensive set of structural data derived from GGA calculations. anions reside. Theoretical calculations have shown that interlayer Sato et al. (2004) and Balan et al. (2005) have also investigated centered interstratification is more favorable than layer centered structural details with ab initio calculations. models (Stixrude and Peacor 2002). For our calculations of elastic In this paper, we begin our calculations of elastic constants properties we adopted the interlayer centered structure model B with the ideal kaolinite structure and compare with the work by of Stixrude and Peacor (2002), with a chemical composition of Sato et al. (2005). To estimate the effects of stacking polytyp- K0.5Al2[Al0.5Si3.5]O10(OH)2, which corresponds to a 1:1 ratio of ism we then compare these with values for three polytypes of illite and smectite, with each layer possessing one aluminum-free kaolinite that have two layers in their unit cell: dickite, nacrite, 3+ tetrahedral sheet and one where 25% of sites are filled by Al and an alternate kaolinite structure. cations. This model differs from the natural smectite mineral Dickite exhibits a 2M stacking sequence, where the vacancy rectorite (e.g., Zhang et al. 2009) in that all interlayer cations in octahedral site alternates between two sites leading to a are assumed to be potassium, as oppose to a mixture of sodium, monoclinic two-layer supercell (Bailey et al. 1988). Such an calcium, and potassium and no interlayer water is included. arrangement leads to increased interlayer binding. According to our DFT calculations, it lowers the energy by 0.01 eV per formula unit relative to the ideal kaolinite structure. For our second example of a two-layer mineral, we chose to look at nacrite because it occurs in nature and its structure has been determined

◄Fi g u r e 3. (left) Illite-smectite model with a chemical composition

of K0.5Al2[Al0.5Si3.5]O10(OH)2, which corresponds to a 1:1 illite-smectite ratio. Each layer is identical and comprises one aluminum-free tetrahedral sheet and one where 25% of sites are fi lled by Al3+ cations. The up-down alternation of these layers leads to two different interlayer environments, one enclosed by low-charge tetrahedral sheets and the other by high-charge tetrahedral sheets, where potassium anions reside. (right) Ideal kaolinite structure where the unit cell has been doubled in →c direction. notation is the same as Figure 1. Militzer et al.: Ab initio Calculation of Elastic Moduli of Sheet Silicates 129 a

c d

Fi g u r e 4. Stacking polytypism in ideal kaolinite (a), dickite (b), nacrite (c), and an additional structure (2M*, d). Only the Si4+ ions (dark) and the Al3+ (light) are shown for simplicity. The view direction is perpendicular to the sheets and a shading has been added to the lower Si4+ layer. The thin lines denote the unit cell. experimentally (Zheng and Bailey 1994; Zhukhlistov 2008). As group tilts into the plane of the layer, preventing it from forming our final example of two-layer stacking, we generated a new a bond with upper layer. Balan et al. (2005) identified this tilt and structure (labeled 2M* throughout this paper) by doubling the explained it as a result of the shorted Si-O bond in nacrite that unit cell of ideal kaolinite, rotating the new layer by 120° and was observed theoretically and experimentally. This tilt was not shifting it by a/3. We selected this particular structure, among reported by Sato et al. (2005) but it was investigated by Benco other possible shifts and rotations, because it shows favorable et al. (2001) who performed finite temperature DFT molecular interlayer hydrogen bonding, which is confirmed by the com- dynamics simulations of kaolinite to compare calculated and puted energy that is again lowered by 0.01 eV per formula unit, measured OH stretching frequencies. relative to ideal kaolinite. The stacking sequence in all four structures is illustrated in Nacrite stands out because it has the largest offset between Figure 4 where, for clarity, we only show the positions of the adjacent layers (Fig. 4), which weakens the hydrogen bonding. tetragonal Si4+ and octahedral Al3+ cations along with the unit cell. The accurate characterization of hydrogen bonds is difficult Studying larger structures would lead to prohibitively expensive experimentally because hydrogen is a weak X-ray scatterer but first-principles calculations. also challenging for density functional theory because of its weak binding forces. Zheng and Bailey (1994) pointed out that Re s u l t s one of the three OH groups may not participate in the interlay- ing bonding and that is what we found when we relaxed the Structural parameters structures with LDA and GGA regardless which experimental For muscovite, we obtained fairly good agreement with geometry we started from. During the relaxation, the outer OH the lattice parameters determined in X-ray diffraction experi- 130 Militzer et al.: Ab initio Calculation of Elastic Moduli of Sheet Silicates

ments (Table 2). Vaughan and Guggenheim (1986) reported: a C33 differ substantially (i.e., LDA: 67.9 and GGA: 19.6 GPa). = 5.1579(9), b = 8.9505(8), c = 20.071(5) Å, and β = 95.75(2)° This reflects the fact that LDA overestimates the density and the P=0 +0.007 (0) while with GGA at ρGGA we find a = 5.238–0.003, b = 9.131 –0.014, interlayer binding while GGA underestimates both. The elastic +0.030 +0.15 c = 20.723–0.045, and β = 95.56–0.65, where we give the results for constants from LDA and GGA computations for the estimated our lowest energy structure and report the deviations from them, density of 2.825 g/cm3 are also compared in Table 3. We consider by the remaining five structures from Table 1 as uncertainty. The that LDA calculations, e.g., C33 = 27.2 GPa, to be our most ac- Si-Al disorder has little impact on the structural parameters, as curate prediction for elastic properties of illite-smectite because, expected. The disagreement with experiment is largest for the for muscovite, GGA overestimated C33 when the experimental lattice parameter c because of the relatively weak binding in this density was used. In general, it will remain challenging for exist- direction. LDA shows better agreement with experiment than ing density functional techniques to predict C33 accurately for GGA. It should be noted that the sample used by Vaughan and weakly bonded sheet silicates in cases where the composition and Guggenheim (1986) exhibited a slightly different composition, structure of the interlayers is variable and poorly defined. K0.9Al2.8Si3.1O12H2.1 compared to the ideal composition that we Tables 4 and 5 summarize the computed elastic constant P=0 adopted. Computations at ρEXP are consequently performed at for the kaolin group minerals. Similar to muscovite, we also P=0 the experimental unit-cell volume rather than the mass density determined values at ρEXP in addition to calculating LDA and but we still refer to as ρP=0 for consistency. The agreement of P =0 P=0 EXP GGA results at ρLDA and ρGGA. Comparison of our calculated the lattice parameter then becomes very good, confirming that elastic constants with those of Sato et al. (2005) reveals sizable the effect on ionic substitution in the Vaughan-Guggenheim discrepancies that are much larger than one would expect from sample is small. For illite-smectite, we cannot directly adopt an experimental Ta b l e 2. Elastic constants of muscovite model 1 in the conventional set- → → density because no explicit experimental data exist for our ideal- ting Z||c calculated with GGA and LDA at different densities ized rectorite model. We thus estimate the density to be 2.825 LDA LDA LDA LDA GGA GGA Experiment 3 g/cm by averaging of the experimental values for the unit-cell PLDA = 0 (a,b,c,β)EXP V = VEXP V = VEXP PGGA = 0 volume of muscovite and pyrophyllite (Table 3). V (Å3) 904.5 917.2 921.9 921.9 921.9 984.3 921.9(3) a (Å) 5.140 5.150 5.1579 5.154 5.187 5.244 5.1579(9) For ideal kaolinite, the calculated lattice parameters in Table b (Å) 8.925 8.945 8.9505 8.952 9.012 9.101 8.9505(8) 4 are in reasonable agreement with the measurements of Neder et c (Å) 19.806 19.998 20.071 20.073 19.818 20.713 20.071(5) al. (1999) although the zero-pressure density is underestimated β (°) 95.48 95.44 95.75 95.43 95.61 95.39 95.75(2) C11 187.5 182.7 180.3 180.9 194.3 170.1 181.0 ± 1.2 by 4% with GGA and overestimated by 4% with LDA. C22 178.1 172.3 169.9 170.0 188.0 162.1 178.4 ± 1.3 Our structural parameters for dickite also slightly deviate C33 71.5 62.7 60.1 60.3 91.1 55.6 58.6 ± 0.6 from the theoretical work of Sato et al. (2004). As seen for ideal C44 22.1 19.9 19.1 18.4 25.2 14.4 16.5 ± 0.6 C55 28.1 25.0 22.4 23.8 30.5 17.8 19.5 ± 0.5 kaolinite in Table 4, we again predict a smaller sheet separation, C66 71.7 71.1 70.6 70.5 71.3 67.6 72.0 ± 0.7 but larger lattice parameters within the sheets. Our value for the C12 59.8 55.1 53.3 53.4 68.1 47.4 48.8 ± 2.5 C 34.8 29.1 27.1 27.2 43.2 20.6 25.6 ± 1.5 angle β of 96.62° is in much better agreement with measured 13 C15 –15.4 –14.7 –14.3 –14.7 –14.3 –14.1 –14.2 ± 0.8 values of 96.48° by Bish and Johnston (1993) and 96.77° by Dera C23 31.0 25.6 23.6 23.5 39.5 18.2 21.2 ± 1.8 et al. (2003) than 104.57° reported by Sato et al. (2004). C25 1.8 1.5 2.3 1.4 1.7 –0.8 1.1 ± 3.7 C35 –0.7 –1.0 0.6 –1.0 1.1 0.0 1.0 ± 0.6 In the case of nacrite, the lattice parameters computed with C46 –0.7 –1.9 –1.8 –1.8 –0.6 –3.3 –5.2 ± 0.9 P=0 GGA at ρGGA are in reasonable agreement with measurements Note: Those calculated with LDA at the experimental unit-cell volume in column by Zhukhlistov et al. (2008): a = 8.910, b = 5.144, c = 14.593 4 (bold) are in best agreement with the experimental values of Vaughan and Guggenheim (1986) in the last column. Å, and β = 100.50° and again the density is underestimated by 4%. It should be noted that, relative to kaolinite, the a and b axis Ta b l e 3. Elastic constants of illite-smectite in the conventional set- → → are swapped. The computed energy is 0.01 eV per formula unit ting Z||c lower than that for ideal kaolinite. LDA PLDA = 0 LDA GGA GGA PGGA = 0 ρ (g/cm3) 2.923 2.8247 2.8247 2.600 Elastic properties a (Å) 5.130 5.156 5.197 5.259 b (Å) 8.861 8.905 8.964 9.066 Table 2 compares our elastic constants calculated for musco- c (Å) 19.738 20.205 19.943 21.116 vite with LDA and GGA with the experimental results of Vaughan α (°) 89.99 89.71 89.92 90.02 and Guggenheim (1986). Elastic constants determined with LDA β (°) 106.08 105.94 106.25 105.74 P =0 γ (°) 89.95 89.94 89.96 89.95 at ρLDA are stiffer than those calculated at the measured density, C11 165.5 153.9 169.1 150.0 P=0 ρEXP, predominantly because LDA overestimates the density. C22 197.8 188.5 202.8 191.2 P=0 C33 67.9 27.2 82.3 19.6 However, elastic constants from LDA calculations at ρEXP are C44 18.6 10.4 10.5 6.9 in very good agreement with experiment. The interlayer spac- C55 35.9 24.8 37.0 19.1 ing, and consequently stiffness, in the direction perpendicular C66 56.3 55.4 56.4 54.3 C12 31.3 25.1 37.4 32.0 to the layers, C33, is most sensitive to changes in density. GGA C13 25.8 13.2 33.4 16.9 yields good agreement with experimental elastic constants, when C15 –23.5 –30.3 –24.4 –36.0 P=0 P=0 C 18.8 5.2 24.9 8.3 calculated at ρGGA, but overestimates values at ρEXP. 23 C –6.9 –8.2 –6.9 –9.6 The computed elastic constants for our illite-smectite model 25 C35 –3.2 –5.4 –0.2 –1.9 are shown in Table 3. Similar predictions are obtained for C11 C46 –15.6 –15.9 –16.7 –15.0 P =0 P=0 Note: Our most reliable results are in bold. and C22 with LDA at ρLDA and GGA at ρGGA, but the values for Militzer et al.: Ab initio Calculation of Elastic Moduli of Sheet Silicates 131 two independent first-principles calculations (Table 4). Sato et coulombic attraction between the negative layers and positive al. (2005) report larger values for C11 and C22, while C33 is less cations, gives rise to strong interlayer bonding. In illite-smectite, than half of our value. only half of the tetrahedral layers have a net negative charge and potassium cations are only present in alternate interlayers, which Di s c u ss i o n leads to weaker interlayer bonding. This explains, for example,

Comparison of our elastic constants for muscovite from why C33 is 50% smaller for illite-smectite than for muscovite. P=0 LDA calculations at ρEXP with those for illite-smectite indicate Differences in other elastic constants are either related to this or that the former is harder than the latter, in qualitative agreement the different concentration and arrangement of defects. with a recent nanoindentation investigation of the two phases Our results for muscovite and illite-smectite form a quantita- by Zhang et al. (2009). The Young’s elastic modulus normal to tive basis for estimating the elastic moduli of the wide variety of the basal plane of muscovite derived from our calculated elastic K-dominated di-octahedral sheet silicate compositions that are constants is 54.7 GPa. This is much lower than the value of 79.3 expected to occur naturally. We propose that the elastic moduli GPa obtained by Zhang et al. (2009), but in good agreement with will scale with the mean inter-layer charge: Z = +1 in the case of values found in earlier investigations, e.g., 58.6 GPa (Vaughan muscovite, and Z = +0.5 in the case of illite-smectite. Smectites and Guggenheim 1986) and 60.9 GPa (McNeil and Grimsditch have values Z < 0.5 and illites have 0.5 < Z < 1.0. Other potentially 1993). For illite-smectite we estimate a value of 26.0 GPa, important effects that remain to be investigated include the role of which is larger than the value of 18.3 GPa obtained by Zhang intra-layer cation radius (e.g., Na substitution for K) and the role et al. (2009). Zhang et al. (2009) suggest that their indenter tip of inter-layer water. The dominant role of inter-layer charge and geometry could have caused their values to be overestimated, local charge balance was also emphasized by Stixrude and Peacor which would explain their larger value for muscovite. For illite- (2002) in their study of the structure of illite-smectite. smetcite, where our value is even larger than that of Zhang et al. The computed elastic constants for kaolinite, dickite, and (2009), we attribute the difference to the fact that their experi- nacrite are summarized in Tables 4–6. These are provided in the → → → → → mental sample was hydrated, whereas we calculated values for conventional (Z||→c,Y||→c×→a, and X = Y×Z) as well as in the more → → → → → → an anhydrous system. intuitive (X||→a,Z||→a×b, and Y = Z×X) orientation. The latter has the → Following similar arguments to those of Zhang et al. (2009) advantage that the Z axis is always oriented perpendicular to the we suggest that the differences in the elastic properties of musco- basal planes and the elastic constants can be compared directly. In → vite and illite-smectite can be related to variations in their crystal the conventional setting, the Z axis changes each time a different structures. In muscovite, all tetrahedral layers have a net negative stacking is introduced. Differences in elastic constants between charge and potassium cations are present in each interlayer. The various structures may, in this setting, be partially due to differ-

Ta b l e 4. Calculated elastic constants for ideal kaolinite in the conven- → → tional setting, Z||c Ta b l e 5. Calculated elastic constants for different kaolin materials using → → Sato et al. This This This This Experiment the conventional setting, Z||c (2004, 2005) work work work work by Neder et Dickite Dickite Nacrite Nacrite Kaolinite Kaolinite al. (1999) GGA LDA GGA LDA model 2M* model 2M* GGA GGA GGA LDA LDA PGGA = 0 ρ = ρEXP PGGA = 0 ρ = ρEXP GGA PGGA = 0 LDA ρ = ρEXP PGGA = 0 ρ = ρEXP ρ = ρEXP PLDA = 0 ρ (g/cm3) 2.505 2.583 2.494 2.607 2.503 2.599 ρ (g/cm3) 2.544 2.506 2.599 2.599 2.711 2.599 Exp. ρ (g/cm3) 2.583 2.583 2.607 2.607 2.599 2.599 a (Å) 5.1445 5.225 5.184 5.179 5.127 5.154 a (Å) 5.221 5.179 9.020 8.910 5.235 5.188 b (Å) 8.9241 9.071 8.999 8.993 8.900 8.942 b (Å) 9.077 9.010 5.221 5.165 9.054 8.973 c (Å) 7.5873 7.464 7.326 7.325 7.182 7.401 c (Å) 14.545 14.323 14.881 14.571 14.582 14.297 α (°) 91.089 91.44 91.73 91.46 91.79 91.69 α (°) 90 90 90 90 96.73 96.81 β (°) 104.60 104.67 105.04 104.66 105.08 104.61 β (°) 96.72 96.62 101.11 101.24 93.32 93.31 γ (°) 89.869 89.77 89.81 89.74 89.80 89.82 γ (°) 90 90 90 90 89.88 89.86

C11 178.5 ± 8.8 166.0 164.1 169.1 169.2 C11 181.1 184.2 147.6 131.8 181.9 184.7 C22 200.9 ± 12.8 177.8 175.5 179.7 178.4 C22 178.6 178.8 160.8 157.9 176.9 185.0 C33 32.1 ± 2.0 70.1 119.3 81.1 149.8 C33 78.6 67.5 64.8 75.0 70.3 81.4 C44 11.2 ± 5.6 13.4 15.3 17.0 19.8 C44 16.9 15.8 7.8 13.2 12.6 16.7 C55 22.2 ± 1.4 21.7 25.2 26.6 29.7 C55 15.5 17.1 8.8 17.7 13.0 16.7 C66 60.1 ± 3.2 56.7 57.4 57.6 58.4 C66 60.6 60.4 62.4 62.2 57.0 57.0 C12 71.5 ± 7.1 64.8 61.2 66.1 65.4 C12 67.7 69.1 45.5 41.5 63.6 65.6 C13 2.0 ± 5.3 16.0 35.1 15.4 41.5 C13 10.8 6.0 5.7 10.2 5.9 6.6 C14 –0.4 ± 2.1 0.0 –0.6 –0.4 –0.2 C14 0.0 0.0 0.0 0.6 –7.5 –7.0 C15 –41.7 ± 1.4 –37.2 –30.5 –34.0 –28.4 C15 –20.3 –17.8 4.4 –16.0 –6.5 -6.9 C16 –2.3 ± 1.7 –7.2 –7.7 –7.8 –8.9 C16 0.0 0.0 0.0 0.1 –4.6 –4.5 C23 –2.9 ± 5.7 11.2 27.2 10.2 33.0 C23 8.0 2.5 12.0 14.0 8.6 9.3 C24 –2.8 ± 2.7 –3.3 –4.7 –3.4 –5.3 C24 0.0 0.0 0.0 0.4 –20.1 –17.9 C25 –19.8 ± 0.6 –16.2 –11.6 –16.1 –10.8 C25 –8.6 –6.3 2.8 –3.8 –2.3 –2.9 C26 1.9 ± 1.5 –0.8 –0.3 –0.1 –0.3 C26 0.0 0.0 0.0 0.1 –1.8 –0.9 C34 –0.2 ± 1.4 –1.3 –0.8 –2.9 –1.2 C34 0.0 0.0 0.0 0.5 3.7 3.2 C35 1.7 ± 1.8 1.3 14.2 6.7 17.0 C35 –2.0 5.8 0.2 9.3 4.8 4.6 C36 3.4 ± 2.2 0.2 0.8 –0.1 0.5 C36 0.0 0.0 0.0 0.1 1.2 1.3 C45 –1.2 ± 1.2 0.1 0.1 –0.7 –0.2 C45 0.0 –0.0 0.0 –0.5 2.4 0.8 C46 –12.9 ± 2.4 –14.2 –13.9 –12.4 –13.0 C46 –6.2 –4.8 0.6 –9.4 –3.7 –3.2 C56 0.8 ± 0.7 0.6 0.5 1.1 1.1 C56 0.0 –0.0 0.0 –0.1 –5.4 –5.1 Note: The experimental density is 2.599 g/cm3. Our most reliable results are Notes: The densities used in the calculations and the experimental values are in bold. listed. Our most reliable results are in bold. 132 Militzer et al.: Ab initio Calculation of Elastic Moduli of Sheet Silicates ences in orientation. Instead of comparing elastic constants, it is (2005) are shorter and agree better with experiment than those of preferable to study pole figures, where a change in orientation only the present work, while the opposite is true for the c parameter. leads to a rotation of the contours. When compared in the same This means that our calculations predict a smaller sheet separa- setting (Table 6), the diagonal elastic constants of ideal kaolinite tion, but larger lattice parameters in the planes. Since slowest agree very well with those of dickite and our 2M* structure. Only wave propagation occurs nearly perpendicular to the planes, the

C33 shows some deviation. Kaolinite deviates from muscovite differences in the structural properties are consistent with Sato and illite-smectite because it is triclinic which leads to additional et al. (2005) predicting a higher elastic anisotropy. non-zero constants, however, most of them are very small with One feature in the pole figures in Figure 5 distinguishes the exception of C16 = −7.8 GPa. Nacrite deviates slightly by kaolin minerals from the others studied here is that the slowest exhibiting a softer response, smaller elastic constants, which is propagation does not occur in the direction perpendicular to the a result of the weaker interlayer hydrogen bonding. layers. Overall, in-plane propagation is still about 50% faster than The measured and calculated elastic constants of muscovite vertical propagation, but there is noticeable asymmetry in the pole lead to very similar predictions for seismic velocities, as the figures. The slowest propagation direction deviates from the plane spherical P wave velocity surfaces in Figure 5 show. The dif- vector by 30 ± 1% for ideal kaolinite, dickite and nacrite, as well ferences are analyzed in more detail in Figure 6 where P- and as the 2M* structure. It is caused by a sizeable contribution from S-wave speeds are plotted for different propagation directions. off-diagonal elastic constants. There is significant shear stress if → → → The direction is changed continuously from Z to X to Y and back the sample is strained vertically. The shear forces are a result of → to Z using equal angular increments. vP varies between 4.5 km/s the hydrogen bonds that are not aligned vertically. in the slow direction perpendicular to the sheets to about 8.3 km/s The results for kaolin minerals in Figure 7 demonstrate shear within the sheets. bands crossing that gives rise to the ring of S wave degeneracy The deviations between experiment and model 1 are no larger in Figure 8. Inside the ring we find a ring of increased splitting than 0.5 km/s, with experimental velocities slightly lower than the magnitude and another minimum inside of that. Conversely mus- calculated one. The results also agree with recent experimental covite, which does not have any hydrogen bonds, does not exhibit studies of muscovite elasticity based on Brillouin scattering (Mc- this unusual shear band signature. Figures 5 and 6 show very little Neil and Grimsditch 1993) and nano-indentation (Zhang et al. shear wave splitting around the vertical direction. This shear band 2009). Just as one expects, splitting of the shear waves is greatest signature makes kaolinite unique among the shale components → → in the X and Y directions, where the polarization is within and studied here. Figure 7 also shows that the elastic constants of Sato → perpendicular to the layers, respectively and lowest in the Z direc- et al. (2004) imply that the P and both S bands are entangled but tion, where both polarizations lie within the plane of the layers. we have no data to support such findings. vP varies between 4.5 km/s in the slow direction perpendicular While the properties of hydrogen bonds have been analyzed to the sheets to about 8.3 km/s within the sheets. in different experimental (e.g., Neder et al. 1999; Dera et al. P-wave velocity surfaces for illite-smectite model B from LDA 2003) and theoretical (e.g., Sato et al. 2004; White et al. 2009) and GGA are shown in Figure 5. The wave speeds parallel to the studies, we show here that they create a distinct signature in sheets are very similar to those of muscovite but perpendicular kaolin minerals that can be measured in laboratory experiments P-velocities are much lower than those for muscovite. The maxi- and seismic observations. → mum in P-velocity of about 8.2 km/s (LDA) occurs along the b direction. The minimum of 2.9 km/s occurs along an axis that is Ta b l e 6. Elastic constants from Tables 4 and 5 computed with LDA at → → → → tilted by 15° away from vertical. the experimental density in the setting X||a→,Z||a→×b, and Y = → → → While Sato et al. do not clearly specify the orientation of their Z×X with the exception of nacrite where we exchanged X and → → → → → → → reported elastic constants, Figure 5 confirms that there is genuine Y (Y||a→, Z||a→×b, and X = Y×Z) to align it in the same way as kaolinite and dickite where the principle axis of the hexagonal disagreement because the velocity isosurfaces differ substantially → lattice in the sheets is parallel to the X direction in shape while a difference in the crystal orientation would only Ideal kaolinite Dickite Nacrite Kaolinite model 2M* lead to a rotation of the isosurfaces. The results of Sato et al. C11 187.4 188.4 136.9 185.3 (2005) predict the structure to be much more anisotropic, with C22 179.8 178.8 157.9 189.3 vP varying between 3.5 and 9.0 km/s, while we predict variations C33 83.6 69.3 79.3 82.5 between 4.9 and 8.8 km/s (Fig. 7). Both sets of calculations in C44 13.7 15.2 11.5 14.2 C55 16.0 14.1 13.0 15.3 Table 4 are based on the Perdew-Berke-Ernzerhof (Perdew et al. C66 61.1 60.9 63.9 58.0 1996) generalized gradient approximation, but use different codes C12 70.5 69.7 42.0 66.6 C 4.8 3.0 5.5 5.0 and pseudopotentials. While Sato et al. used the CASTEP program 13 C14 –0.5 0.0 0.8 –0.0 and ultrasoft pseudopotentials with a 340 eV plane wave energy C15 0.2 –0.1 3.6 1.1 cut-off, we used the VASP code and the projector augmented- C16 –7.5 0.0 –0.0 –3.7 C23 6.0 2.0 13.6 6.8 wave method with a 586 eV cut-off or higher. The latter is more C24 0.4 0.0 0.5 0.0 reliable than ultrasoft pseudopotentials and is expected to yield C25 –0.3 1.5 1.7 0.5 results that are closer to all-electron calculations. We also notice C26 0.8 0.0 –0.0 0.3 C34 –4.3 0.0 0.4 –2.1 some differences in the reported structural parameters given in C35 –2.6 1.9 1.1 2.0 Table 4 along with the experimental values by Neder et al. (1999). C36 0.7 0.0 –0.0 0.1 C –0.2 0.0 –0.5 –0.4 The angles from both calculations agree well with experiments. 45 C46 –0.9 0.4 0.6 –0.9 On the other hand, lattice parameters a and b from Sato et al. C56 0.2 0.0 0.0 –0.4 Militzer et al.: Ab initio Calculation of Elastic Moduli of Sheet Silicates 133

8 7.5 7 6.5 6 5.5 5 4.5 km/s

Fi g u r e 5. P wave velocity surfaces derived from elastic constants in Tables 2–5: Muscovite computed using model 1 and experimental results from Vaughan and Guggenheim (1986), illite-smectite comparison of LDA and GGA for ρ = 2.825 g/cm3, ideal kaolinite from Sato et al. and this → P=0 work, and finally our results for dickite, nacrite and our M2 * stacking model, all calculated at ρEXP using LDA. Equal area projections along the Z → → = [001] direction in the conventional setting, Z||c, is used throughout.

acoustic wave velocities are much lower than those computed 8 from matrix averaging. Is this difference due to reduced elastic properties of sheet silicates, such as illite-smectite, or is it caused by the pore structure that is not taken into account for the averag- 7 Experiment ing? This was the motivation behind this study. vp Model 1 To demonstrate the application of the computed single-crystal 6 properties to natural shales, we have chosen a well-characterized shale, composed mainly of detrital and authigenic illite, kaolinite, 5 and quartz, with minor plagioclase, pyrite, and chlorite. The shale is of Kimmeridgian age (156–151 Ma) and from a drill core in vs1 4 the North Sea. Elastic properties were determined as a function of pressure with ultrasonic methods (Hornby 1998). The same Elastic wave speed (km/s) sample was recently reinvestigated with synchrotron X-ray dif- 3 v s2 fraction to quantify orientation distributions of component phases (Wenk et al. 2010). The Kimmeridge Clay shale shows strong 2 ZXYZpreferred orientation of kaolinite, illite, and illite-smectite, and Propagation direction a nearly random orientation distribution for quartz. The pole figures in Figure 9 display nearly axial symmetry of orientation Fi g u r e 6. Elastic wave velocities for muscovite disorder model 1 distributions. Therefore, strict axial symmetry was imposed in the (Fig. 2) from Table 2 column 4 and experimental results from Vaughan following property calculations (transverse isotropy), reducing and Guggenheim (1986) as function of propagation direction using the → → the number of independent components of the aggregate elastic conventional setting, Z||c. tensor to five: C11 = C22, C33, C12 = C11 − 2C66, C13 = C23, C14 =

C55 and all others are zero. To estimate the elastic properties of the shale, first the elastic Application to shale anisotropy properties of contributing mineral phases were calculated by An important application of improved single-crystal elastic averaging their single-crystal values over the mineral orientation properties of clay minerals is to polycrystal averages for shales. distributions, using a geometric mean (Matthies and Humbert Shales are complex rocks, rich in sheet silicates. To estimate the 1993) in the software Beartex (Wenk et al. 1998). For single- P=0 contribution of the matrix to elastic properties, and particularly crystal properties we used the LDA values at ρLDA for illite and elastic anisotropy, we need to know volume fractions of constitu- illite-smectite, and for kaolinite the LDA values for kaolinite. For ent minerals, their orientation distributions and single-crystal quartz we used the experimental data of Heyliger et al. (2002). elastic properties. It has previously been observed that measured It should be mentioned that in these texture-related calculations 134 Militzer et al.: Ab initio Calculation of Elastic Moduli of Sheet Silicates

Hornby (1998) measured at the maximum confining pressure of 9 80 MPa. In the bulk average, the porosity, estimated at 2.5%, was 8 not taken into account. From aggregate elastic properties, P- and S-wave velocities were calculated and Figure 10 shows P-veloci- 7 ties and shearwave splitting as function of the angle to the bed- 6 ding plane. Illite and illite-smectite are strongly anisotropic (32 and 30%, respectively), combining strong preferred orientation 5 and strong single-crystal anisotropy. The anisotropy of kaolinite 4 is smaller because of reduced single-crystal anisotropy (16%) and the contribution of quartz to anisotropy is negligible. 3 Note that using the new single-crystal elastic constants, the calculated values for the shale aggregate are still considerably 2 Elastic wave speed (km/s) Ideal koalinite higher than experimental results in Table 7 but the differences are This work 1 Nacrite less than in previous comparisons (Voltolini et al. 2009; Wenk et Sato et al. al. 2008). Particularly, using the illite-smectite results instead of 0 ZXYZmuscovite reduces the value of C33 from 85.0 to 70.5 GPa, which is still much higher than the measured value of 36 GPa. Propagation direction The discrepancy between measured and computed aggregate elastic moduli is likely due to the pore/fracture structure of the Fi g u r e 7. Elastic wave speeds compared for ideal kaolinite and rock. There is an extensive literature that discusses the influence nacrite. The upper branch shows vP and the lower two show vS. of pores (e.g., Bayuk et al. 2007; Berryman et al. 2002; Mukerji a consistent approach must be used for phases with monoclinic et al. 1995), fracture distribution (Sayers 1998) and saturation crystal symmetry. Beartex uses the first setting ([001] twofold (Pham et al. 2002) on the elastic properties of shales. The ap- axis) and corresponding transformations of lattice parameters, proach of the models is generally empirical since very little is pole figure indices, and elastic constants are required (Matthies known about the details of the pore structure and distribution and and Wenk 2009). In this paper we represent data in second does not consider anisotropy. This may change in the future as setting ([010] twofold axis), even though all calculations were micro- and nano-tomography techniques are becoming available done in the first setting. For the bulk aggregate elastic constants to map the 3D pore structure (e.g., Bleuet et al. 2008; Herman of the contributing phases were averaged, taking into account 2009; Kanitpanyacharoen et al. 2011). Recently Matthies (2010) corresponding volume fractions. proposed a self-consistent model based on the theory of Eshelby Table 7 lists aggregate elastic constants of the contributing (1957) to include penny- or rod-shaped pores in a heterogeneous phases, as well as the bulk average and experimental data from anisotropic medium and this could prove valuable for further

Z=[001]

Fi g u r e 8. Comparison of shear wave splitting in muscovite and ideal kaolinite. The latter exhibits a ring of nearly perfect degeneracy in shear wave velocity. The dashes indicate the direction of polarization of the fast shear wave. Militzer et al.: Ab initio Calculation of Elastic Moduli of Sheet Silicates 135

Ta b l e 7. Elastic constants for Hornby Kimmeridge shale (in GPa)

Phase vol% C11 C33 C44 C12 C13 P velocity anisotropy (%) Illite-mica 26.7 118.2 81.2 30.6 36.8 33.3 18.7 Illite-smectite 45.5 83.6 49 22.6 20.6 19.3 26.7 Kaolinite 5.1 121.7 87.3 29.1 43.6 34.5 16.6 Quartz 22.7 97.3 97.1 43.9 8.1 7.8 0 Average 97.9 70.5 29.9 23.3 21.2 16.5 Experiment 56.2 36.4 10.3 18.4 20.5 21.6

a 7

6 m/s)

locities (k P ve Quartz 4 lllite l/S Kaolinite Average Exp. 3 Fi g u r e 9. (001) pole figures for detrital illite, illite-smectite, 0 10 20 30 40 50 60 70 80 90 kaolinite, and quartz in equal-area projections on the bedding plane. Angle from bedding normal (deg.) Contours are in multiples of a random distribution.

b 700 quantifying elastic properties of shales. The important role of porosity in the elasticity of the experimental sample is further 600 emphasized by its variation with confining pressure, which re- 500 duces the anisotropy substantially due to closing of pores and/ or cracks (Hornby 1998). 400 Even without a quantitative model the comparison of measured and calculated elastic properties reveals important elocities (m/s) 300 Quartz lllite information about the pore structure in Kimmeridge Clay shale. ds v 200 l/S Since the relative difference between computed and measured Kaolinite Average aggregate moduli is much greater for C33 than for C11, pores must 100 Exp. have very anisotropic shapes and alignment. The pattern suggests 0 pores that are flattened and aligned in the bedding plane. The 0 10 20 30 40 50 60 70 80 90 difference between computed and measured Cij could be used in Angle from bedding normal (deg.) principle to invert for the properties of a simple pore geometry model, such as one based on penny-shaped cracks, although this Fi g u r e 10. P-velocities (a) and shear wave splitting (b) vs. angle to is beyond the scope of the present study. In the future we plan to bedding plane normal. The computed results for the components were investigate this shale by micro-tomography and use this informa- averaged to compare with the experimental values for the aggregate tion to average over both pores and matrix minerals. Combined (Hornby 1998). with our new values of single-crystal elastic moduli we hope to obtain more reliable estimates of shale elasticity. In summary, we have calculated the single-crystal elastic (dickite) are expected to be stiffer than those where it is weak properties of the layered silicates muscovite, illite-smectite, (nacrite). The calculated elastic properties of sheet silicate kaolinite, dickite, and nacrite, from density functional theory. minerals provide a quantitative basis on which to discuss the Our results suggest that, for dioctahedral sheet silicates, cation elasticity of shales, which should be of great benefit for seismic disorder has little effect on the elastic properties of individual prospecting of hydrocarbon deposits. aluminosilicate layers. For muscovite calculated elastic properties are in excellent agreement with experiments. The elastic Ac k n o w l e d g m e n t s properties of naturally occurring clays depend on the degree and B.M. and S.S. acknowledge support from NSF (CMG 0530282) and U.C. type of isomorphic substitution and interlayer cations, since it Berkeley’s lab fee grant program. Teragrid and NCCS computers were used. controls the strength of interlayer bonding. Those with higher H.R.W. acknowledges support from DOE-BES (DE-FG02-05ER15637), NSF (EAR-0337006), and the Esper Larsen Fund. We acknowledge access to the mean inter-layer charge (e.g., illites) are expected to be stiffer facilities of beamline 11-ID-C at APS ANL for texture measurements and Y. Ren then those with lower inter-layer charge (e.g., illite-smectites). for assistance with the experiments. Brian Hornby (BP) kindly provided us with a In a similar manner, in the case of kaolin minerals, stacking sample of Kimmeridge Clay shale. We appreciate constructive reviews by Nico de Koker and Manuel Sintubin that helped us improve the manuscript. arrangements that result in strong interlayer hydrogen bonding 136 Militzer et al.: Ab initio Calculation of Elastic Moduli of Sheet Silicates

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